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Function defined by a hypergeometric series
ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as
Hypergeometric_function
Family of power series in mathematics
generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n {\displaystyle n} is a rational function of n
Generalized hypergeometric function
Generalized_hypergeometric_function
Solution of a confluent hypergeometric equation
a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential
Confluent hypergeometric function
Confluent_hypergeometric_function
Q-analog of hypergeometric series
by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the
Basic_hypergeometric_series
Hypergeometric function in mathematics
mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced
General hypergeometric function
General_hypergeometric_function
Sigmoid shape special function
Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function): erf ( x ) = 2 x π M ( 1 2 , 3 2 , − x 2 ) . {\displaystyle
Error_function
Elliptic analog of hypergeometric series
is an elliptic function of n {\displaystyle n} , analogous to generalized hypergeometric series where the ratio is a rational function of n {\displaystyle
Elliptic hypergeometric series
Elliptic_hypergeometric_series
Extension of the factorial function
functions can be expressed in terms of the gamma function. More functions yet, including the hypergeometric function and special cases thereof, can be represented
Gamma_function
Mathematical function
{s^{b-1}}{(1+s)^{a+b}}}\,ds.} It can also be written in terms of the hypergeometric function: B ( x ; a , b ) = x a a 2 F 1 ( a , 1 − b ; a + 1 ; x ) {\displaystyle
Beta_function
mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an
Hypergeometric function of a matrix argument
Hypergeometric_function_of_a_matrix_argument
Generalization of the hypergeometric function
of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's G-function was able to include those as
Meijer_G-function
Multivalued function in mathematics
generalization resembles the hypergeometric function and the Meijer G function but it belongs to a different class of functions. When r1 = r2, both sides
Lambert_W_function
Family of solutions to related differential equations
}e^{-x\sinh t-\alpha t}\,dt.} The Bessel functions can be expressed in terms of the generalized hypergeometric series as J α ( x ) = ( x 2 ) α Γ ( α +
Bessel_function
Topics referred to by the same term
Hypergeometric may refer to several distinct concepts within mathematics: The hypergeometric function, a solution to the Gaussian hypergeometric differential
Hypergeometric
Discrete probability distribution
random variable X {\displaystyle X} follows the hypergeometric distribution if its probability mass function (pmf) is given by p X ( k ) = Pr ( X = k ) =
Hypergeometric_distribution
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Types of special mathematical functions
{z^{s+k}}{s+k}}={\frac {z^{s}}{s}}M(s,s+1,-z),} where M is Kummer's confluent hypergeometric function. When the real part of z is positive, γ ( s , z ) = s − 1 z s e
Incomplete_gamma_function
Generalization of the Meijer G-function and the Fox–Wright function
Sena Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions". Researchgate. Retrieved 1 March 2023. Mathai, A
Fox_H-function
Mathematical theorem on convolved binomial coefficients
case of Gauss's hypergeometric theorem, which states that 2F1(a,b;c;1)=Γ(c)Γ(c−a−b)Γ(c−a)Γ(c−b) where 2F1 is the hypergeometric function and Γ(n+1)=n! is
Vandermonde's_identity
Sequence of differential equation solutions
generating function for them is ∑n=0∞tnLn(α)(x)=1(1−t)α+1e−tx/(1−t). Laguerre functions are defined by confluent hypergeometric functions and Kummer's
Laguerre_polynomials
In physics, solution to Schrödinger equation
potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument. The Coulomb wave equation for
Coulomb_wave_function
Polynomial sequence
Gustav Jacob Jacobi. The Jacobi polynomials are defined via the hypergeometric function as follows: P n ( α , β ) ( z ) = ( α + 1 ) n n ! 2 F 1 ( − n
Jacobi_polynomials
Generalisation of the generalised hypergeometric function pFq(z)
function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric
Fox–Wright_function
Special function in the physical sciences
mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after
Airy_function
Probability distribution
the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n
Binomial_distribution
Polynomial sequence
hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions
Hermite_polynomials
Anger–Weber function Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral Paul Émile Appell (1855–1930): Appell hypergeometric series
List of eponyms of special functions
List_of_eponyms_of_special_functions
In mathematics, a solution to a modified form of the confluent hypergeometric equation
mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by
Whittaker_function
Mathematical functions
are increasingly popular. In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and
Falling_and_rising_factorials
In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman
Bateman_function
function Riesz function Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions Meijer G-function
List of mathematical functions
List_of_mathematical_functions
Polynomial sequence
Chebyshev polynomials of the second kind. They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite: C n ( α
Gegenbauer_polynomials
Set of four hypergeometric series
of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable
Appell_series
Type of mathematical series
two consecutive terms is a rational function of n {\displaystyle n} . The definition of the generalized hypergeometric series is similar, except that the
Bilateral hypergeometric series
Bilateral_hypergeometric_series
Solutions of Legendre's differential equation
converted into a hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions. Since the
Legendre_function
Input to a mathematical function
hypergeometric function is an example of a four-argument function. The number of arguments that a function takes is called the arity of the function.
Argument_of_a_function
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
of hypergeometric identities. Hypergeometric function lists identities for the Gaussian hypergeometric function Generalized hypergeometric function lists
List of hypergeometric identities
List_of_hypergeometric_identities
Mathematical function having a characteristic S-shaped curve or sigmoid curve
functions M25: Special cases of Gauss hypergeometric functions M26: Feedback closed-loop systems M27: Recursive functions M28: Recursive time-delayed feed-forward
Sigmoid_function
Contour integral involving a product of gamma functions
product of gamma functions. They were introduced by Ernest William Barnes (1908, 1910). They are closely related to generalized hypergeometric series. The
Barnes_integral
Number of subsets of a given size
\alpha } . Binomial transform Delannoy number Eulerian number Hypergeometric function List of factorial and binomial topics Macaulay representation of
Binomial_coefficient
Special function defined by an integral
connexion with the confluent hypergeometric functions is that E 1 {\displaystyle E_{1}} is an exponential times the function U ( 1 , 1 , z ) {\displaystyle
Exponential_integral
Canonical solutions of the general Legendre equation
gamma function and 2F1 is the hypergeometric function 2F1(α,β;γ;z)=Γ(γ)Γ(α)Γ(β)∑n=0∞Γ(n+α)Γ(n+β)Γ(n+γ) n!zn, They are called the Legendre functions when
Associated Legendre polynomials
Associated_Legendre_polynomials
Special function defined by an integral
{i^{k}}{(m+nk+1)}}{\frac {x^{m+nk+1}}{k!}}} is a confluent hypergeometric function and also an incomplete gamma function ∫ x m e i x n d x = x m + 1 m + 1 1 F 1 ( m
Fresnel_integral
Pair of polynomial sequences
yieldsTn(x)=∑j=0⌊n/2⌋(n2j)(x2−1)jxn−2j.This can be written as a 2F1 hypergeometric function: Tn(x)=∑k=0⌊n/2⌋(n2k)(x2−1)kxn−2k=xn∑k=0⌊n/2⌋(n2k)(1−x−2)k=12n
Chebyshev_polynomials
Pair of functions in combinatorics
involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent and much
Wilf–Zeilberger_pair
Family of orthogonal polynomials
polynomials. Hahn polynomials are defined in terms of generalized hypergeometric functions by Qn(x;α,β,N)=3F2(−n,−x,n+α+β+1;α+1,−N+1;1) for n=0,1,…,N. If
Hahn_polynomials
–2t/(1–t2) An explicit expression for them in terms of the generalized hypergeometric function 3F0: s n ( x ) = ( − x / 2 ) n 3 F 0 ( − n , 1 − n 2 , 1 − n 2
Mott_polynomials
Well defined hypergeometric series discovered by Giuseppe Lauricella
In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables. They are (Lauricella 1893): F A ( 3 ) ( a
Lauricella hypergeometric series
Lauricella_hypergeometric_series
Analytic function that does not satisfy a polynomial equation
generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for some special parameter values. Transcendental functions cannot
Transcendental_function
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Summation method for hypergeometric terms
where S(n) is a hypergeometric term (i.e., S(n + 1)/S(n) is a rational function of n); then necessarily a(n) is itself a hypergeometric term, and given
Gosper's_algorithm
System of complete and orthogonal polynomials
polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions. In this approach,
Legendre_polynomials
Japanese mathematician
is a Japanese mathematician who introduced the Aomoto-Gel'fand hypergeometric function and the Aomoto integral. He was a professor at Nagoya University
Kazuhiko_Aomoto
Concept in mathematics
) {\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)} is the confluent hypergeometric function. Other pairs of independent solutions may be formed from linear
Parabolic_cylinder_function
Special mathematical functions defined on the surface of a sphere
group is given by the hypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as SO(3) = PSU(2)
Spherical_harmonics
generalized hypergeometric type, and in fact the Bessel–Clifford function is up to a scaling factor a Pochhammer–Barnes hypergeometric function; we have
Bessel–Clifford_function
Irreducible representation of the rotation group SO
) s i m − m ′ , {\displaystyle (-1)^{s}i^{m-m'},} causing half of the functions to be purely imaginary. The realness of the d-matrix elements is one of
Wigner_D-matrix
Discrete probability distribution
special case where α and β are integers is also known as the negative hypergeometric distribution. The beta distribution is a conjugate distribution of the
Beta-binomial_distribution
Mathematical identities related to integer partitions
the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered
Rogers–Ramanujan_identities
Special function defined by an integral
In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number
Logarithmic_integral_function
Probability distribution
characteristic function of the beta distribution to a Bessel function, since in the special case α + β = 2α the confluent hypergeometric function (of the first
Beta_distribution
In mathematics, the E-function was introduced by Thomas Murray MacRobert (1937–1938) to extend the generalized hypergeometric series pFq(·) to the case
MacRobert_E_function
German polymath and scholar (1777–1855)
general hypergeometric function F(α,β,γ,x), and shows that many of the functions known at the time are special cases of the hypergeometric function. This
Carl_Friedrich_Gauss
Generalization of the hypergeometric differential equation
equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur
Riemann's differential equation
Riemann's_differential_equation
Special function in mathematics
In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de
Kampé_de_Fériet_function
Probability distribution
Distribution". Wroughton, Jacqueline. "Distinguishing Between Binomial, Hypergeometric and Negative Binomial Distributions" (PDF). Hilbe, Joseph M. (2011)
Negative binomial distribution
Negative_binomial_distribution
Type of functions, in mathematical analysis
the class of hypergeometric functions. Examples of special functions that are holonomic but not hypergeometric include the Heun functions. Examples of
Holonomic_function
Special functions used to build correlation functions in 2D CFTs
the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated. In
Virasoro_conformal_block
Topics referred to by the same term
G-function, related to the Gamma function Meijer G-function, a generalization of the hypergeometric function Siegel G-function, a class of functions in
G-function
Special function defined by an integral
the sinc function, and also the zeroth spherical Bessel function. Since sinc {\displaystyle \operatorname {sinc} } is an even entire function (holomorphic
Trigonometric_integral
Probability distribution
particular instance of the hypergeometric function. For information on its inverse cumulative distribution function, see quantile function § Student's t-distribution
Student's_t-distribution
Measure of linear correlation
is the gamma function and 2 F 1 ( a , b ; c ; z ) {\displaystyle {}_{2}\mathrm {F} _{1}(a,b;c;z)} is the Gaussian hypergeometric function. In the special
Pearson correlation coefficient
Pearson_correlation_coefficient
Real root of the polynomial x^5+x+a
series for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The equation x 5 + x + a = 0 {\displaystyle
Bring_radical
Risk measure estimating the average loss in the worst tail of the distribution
beta function is defined only for positive arguments, for a more generic case the expected shortfall can be expressed with the hypergeometric function: ES
Expected_shortfall
Orthogonal polynomials
Carl Charlier in 1905. They are given in terms of the generalized hypergeometric function by C n ( x ; μ ) = 2 F 0 ( − n , − x ; − ; − 1 / μ ) = ( − 1 )
Charlier_polynomials
Linear recurrence equation
quotient of two polynomials. A hypergeometric function f {\textstyle f} is a function in the broad class of analytc functions that can be represented as a
P-recursive_equation
Discrete probability distribution
In probability theory and statistics, the negative hypergeometric distribution describes probabilities for when sampling from a finite population without
Negative hypergeometric distribution
Negative_hypergeometric_distribution
Type of function in mathematics
special functions are analytic on a suitable domain: hypergeometric functions on suitable domains Bessel functions on suitable domains The gamma function away
Analytic_function
Triangular array of the binomial coefficients
convolving the distribution function for a random variable with itself corresponds to calculating the distribution function for a sum of n independent
Pascal's_triangle
Recurrence relations of binomial coefficients in Pascal's triangle
distribution · Hypergeometric distribution · Poisson distribution · Gamma distribution Special functions (connections) Gamma function · Beta function · Incomplete
Hockey-stick_identity
functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function ϕ {\displaystyle \phi } by J ν ( 1 ) ( x ; q ) = ( q ν + 1 ; q
Jackson_q-Bessel_function
In mathematics, the Toronto function is a modification of the confluent hypergeometric function defined by A. H. Heatley as T(m,n
Toronto_function
Special function defined by an integral
where n!! denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed
Elliptic_integral
Algebraic expansion of powers of a binomial
{n}{k}}={\frac {n!}{k!\;(n-k)!}},} which is defined in terms of the factorial function n!. Equivalently, this formula can be written ( n k ) = n ( n − 1 ) ⋯ (
Binomial_theorem
q\right]} where φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral
Askey–Wilson_polynomials
Type of permutation of a set of elements
Dn=[n!e]=⌊n!e+12⌋for n≥1, where [x] is the nearest integer function and ⌊x⌋ is the floor function. Other related formulas include Dn=⌊n!+1e⌋for n≥1,Dn=⌊(e+e−1)n
Derangement
Monochrome light beam whose amplitude envelope is a Gaussian function
real-valued, Γ(x) is the gamma function and 1F1(a, b; x) is a confluent hypergeometric function. Some subfamilies of hypergeometric-Gaussian (HyGG) modes can
Gaussian_beam
Product of consecutive factorial numbers
continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function as s f ( n ) = G ( n + 2 ) {\displaystyle
Superfactorial
Equalities involving sums over the coefficients occurring in hypergeometric series
mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These
Hypergeometric_identity
Measure of internal forces in an atomic nucleus
}{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}} is the hypergeometric function. It is also possible to analytically solve the eigenvalue problem
Woods–Saxon_potential
In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions. They are named after Norman Macleod Ferrers
Ferrers_function
Mathematical equation
the form of the hypergeometric differential equation. It has two linearly independent solutions, called the periods of elliptic functions. The ratio of
Picard–Fuchs_equation
Generalization of the binomial theorem to other polynomials
Kummer's theorem. By Stirling's approximation, or equivalently the log-gamma function's asymptotic expansion, log ( k n n , n , ⋯ , n ) = k n log ( k ) +
Multinomial_theorem
special functions, Schwarz's list or the Schwarz table is the list of 15 cases found by Hermann Schwarz (1873, p. 323) when hypergeometric functions can be
Schwarz's_list
Cunningham (1908). It can be defined in terms of the confluent hypergeometric function U, by ω m , n ( x ) = e − x + π i ( m / 2 − n ) Γ ( 1 + n − m /
Cunningham_function
theory of special functions in mathematics, the Horn functions (named for Jakob Horn) are the 34 distinct convergent hypergeometric series of order two
Horn_function
their properties. The polynomials are given in terms of basic hypergeometric functions by H n ( x | q ) = e i n θ 2 ϕ 0 [ q − n , 0 − ; q , q n e − 2
Continuous q-Hermite polynomials
Continuous_q-Hermite_polynomials
Russian mathematician (born 1962)
investigated generalized Euler integrals, A {\displaystyle A} -hypergeometric functions, A {\displaystyle A} -discriminants, and hyperdeterminants, and
Mikhail_Kapranov
Transformation of a mathematical sequence
R. B. (2010). "Euler-type transformations for the generalized hypergeometric function". Z. Angew. Math. Phys. 62 (1): 31–45. doi:10.1007/s00033-010-0085-0
Binomial_transform
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